OFFSET
1,1
COMMENTS
Numbers n such that P(phi(n)) - phi(P(n)) = 1, where P(x) is the largest prime factor of x. P(phi(n)) - phi(P(n)) = A006530(A000010(n)) - A000010(A006530(n)).
Numbers n such that the value of the commutator of phi and P functions at n is -1.
Equivalently, n such that n and phi(n) have the same largest prime factor since Phi(p) = p-1 if p is prime. - Benoit Cloitre, Jun 08 2002
Since n is divisible by P(n)^2, n cannot divide P(n)! and so A057109 is a supersequence. Hence all A002034(a(n)) are composite. - Jonathan Sondow, Dec 28 2004
A225546 defines a self-inverse bijection between this sequence and A335740, considered as sets. - Peter Munn, Jul 19 2020
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Paul Erdős and Ron L. Graham, On products of factorials, Bull. Inst. Math. Acad. Sinica 4:2 (1976), pp. 337-355. [alternate link]
Paul Erdős and Ilias Kastanas, Solution 6674:The smallest factorial that is a multiple of n, Amer. Math. Monthly 101 (1994) 179.
A. J. Kempner, Miscellanea, Amer. Math. Monthly, 25 (1918), 201-210. See Section II, "Concerning the smallest integer m! divisible by a given integer n."
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Eric Weisstein's World of Mathematics, Totient Function
FORMULA
Erdős proved that there are x * exp(-(1 + o(1))sqrt(log x log log x)) members of this sequence up to x. - Charles R Greathouse IV, Mar 26 2012
MAPLE
isA070003 := proc(n)
if modp(n, A006530(n)^2) = 0 then # code re-use
true;
else
false;
end if;
end proc:
A070003 := proc(n)
option remember ;
if n =1 then
4;
else
for a from procname(n-1)+1 do
if isA070003(a) then
return a
end if;
end do:
end if;
end proc:
seq( A070003(n), n=1..80) ; # R. J. Mathar, Jun 27 2024
MATHEMATICA
p[n_] := FactorInteger[n][[-1, 1]]; ep[n_] := EulerPhi[n]; fQ[n_] := p[ep[n]] == 1 + ep[p[n]]; Select[ Range[ 510], fQ] (* Robert G. Wilson v, Mar 26 2012 *)
Select[Range[500], FactorInteger[#][[-1, 2]] > 1 &] (* T. D. Noe, Dec 06 2012 *)
PROG
(PARI) for(n=3, 1000, if(component(component(factor(n), 1), omega(n))==component(component(factor(eulerphi(n)), 1), omega(eulerphi(n))), print1(n, ", ")))
(PARI) is(n)=my(f=factor(n)[, 2]); f[#f]>1 \\ Charles R Greathouse IV, Mar 21 2012
(PARI) sm(lim, mx)=if(mx==2, return(vector(log(lim+.5)\log(2)+1, i, 1<<(i-1)))); my(v=[1]); forprime(p=2, min(mx, lim), v=concat(v, p*sm(lim\p, p))); vecsort(v)
list(lim)=my(v=[]); forprime(p=2, sqrt(lim), v=concat(v, p^2*sm(lim\p^2, p))); vecsort(v) \\ Charles R Greathouse IV, Mar 27 2012
(Python)
from sympy import factorint
def ok(n): f = factorint(n); return f[max(f)] >= 2
print(list(filter(ok, range(4, 508)))) # Michael S. Branicky, Apr 08 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, May 07 2002
EXTENSIONS
New name from Jonathan Sondow and Charles R Greathouse IV, Mar 27 2012
STATUS
approved